Optimal. Leaf size=175 \[ -\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{3/2}}+\frac {(69 x+241) \sqrt {3 x^2+5 x+2}}{10 \sqrt {2 x+3}}+\frac {289 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{4 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {367 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{20 \sqrt {3 x^2+5 x+2}} \]
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Rubi [A] time = 0.10, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {812, 843, 718, 424, 419} \[ -\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{3/2}}+\frac {(69 x+241) \sqrt {3 x^2+5 x+2}}{10 \sqrt {2 x+3}}+\frac {289 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{4 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {367 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{20 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{5/2}} \, dx &=-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^{3/2}}-\frac {1}{10} \int \frac {(-173-207 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx\\ &=\frac {(241+69 x) \sqrt {2+5 x+3 x^2}}{10 \sqrt {3+2 x}}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^{3/2}}+\frac {1}{60} \int \frac {-2787-3303 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {(241+69 x) \sqrt {2+5 x+3 x^2}}{10 \sqrt {3+2 x}}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^{3/2}}-\frac {1101}{40} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {289}{8} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {(241+69 x) \sqrt {2+5 x+3 x^2}}{10 \sqrt {3+2 x}}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^{3/2}}+\frac {\left (289 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{4 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (367 \sqrt {3} \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{20 \sqrt {2+5 x+3 x^2}}\\ &=\frac {(241+69 x) \sqrt {2+5 x+3 x^2}}{10 \sqrt {3+2 x}}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^{3/2}}-\frac {367 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{20 \sqrt {2+5 x+3 x^2}}+\frac {289 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{4 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 195, normalized size = 1.11 \[ -\frac {2 \left (54 x^5-396 x^4+777 x^3+6107 x^2+7444 x-117 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt {\frac {3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+2564\right )+1101 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{5/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{60 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 213, normalized size = 1.22 \[ \frac {-1080 x^{5}+7920 x^{4}+116580 x^{3}+296240 x^{2}+2202 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+688 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+269500 x +3303 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1032 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+80840}{600 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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